What are the odds of …. ?

My mind is kind of blown right now. Need to put this into words and write it down. Perhaps others have some input to validate or contradict what I am saying.

I was studying probabilities and going through a topic of binomial distributions. I finished the lectures and went online to see how this information could be useful in the real world. Textbook examples are all about coin flips and I had a hard time relating the information to the real world scenarios. How often do you need to know the probability of tails coming up twice in three flips? Why does this binomial distribution topic matter?

A cursory search brought more coin examples. The real stuff was starting to show through here and there, starting with this article. Yet nothing that I could relate to or apply directly. Until it hit me: binomial distribution could help explain some of the NBA betting behavior I have been seeing! Let me backtrack to how I got there.

Background

I’ve been running an experiment to see if it would be possible to come out cash positive when betting on NBA games using wagering lines provided online. A short answer to that is a “no”: gambling institutions are sophisticated money-making operations that have no interest in losing money to a common man. Yet, the temptation to find an edge is strong, if you combine that with machine learning and stats analysis, one can’t help but try.

For three months now I simulate NBA bets against the real lines and record what happens. The first two months resulted in a negative balance. Last month+ has been positive. My initial approach that was losing money was strictly using a machine learning model I built and I made bets using moneyline odds. It was a good experience in that I learned that favorites in NBA lose much more often than a casual fan realizes. I also learned a ton on how simple ML models won’t cut it against skewed odds that oddsmakers setup against you, simple stab won’t work.

Bad results did not discourage me from experimenting, and I decided to see what would happen if I used the model only as one of the signals, and then relied on my own intuition as I watch the games and keep up with the teams. I no longer did “moneylines” and instead picked “the spread” bets that have payoff anywhere from -115 to -105. I also decided to only bet up to three games per day regardless of the number of matchups in the day. The three game choice was random, but I figured it was a number that forced me to think about the choices harder and pick the games I had the most confidence. With the spread odds, picking two out of three correctly guaranteed a positive night if you bet equal amounts on each game. 66% target rate did not sound too intimidating.

The new strategy has been in place for most of December and all of January. During this time I am winning ~62% of bets and slowly making up for the losses encountered during the first two months. 62% is close to 66% I was hoping to achieve. I track each bet, and the average odds come out to be ~-114. Basically for $10, you win $8.77 ($10 * 1000 / 114). Expected Value with params like this is positive: 0.62 * 8.77 – 0.38 * 10 = 5.43 – 3.8 = 1.63. All of this is assuring. I still can’t explain the winning rate with formula since intuition is a big part of it. This is concerning since I could be just hitting a lucky streak but for now, I am continuing with the approach.

During all of the time that the strategy has been running, I’ve had two questions in the back of my mind. My most common outcome when betting three games has been two wins and one loss. Kind of makes sense since in aggregate I am hitting about 62% win rate. But I wondered “what are the odds of hitting all three correctly? what about two, and one, and zero?”. And also wondered if I should continue to make three bets a day or increase it. I did not spend too much time trying to answer those questions and instead focused on getting better data, etc.

Aha! moment

With all the background info out of the way, here is the big realization of the day: betting on games is exactly where the field of binomial distribution can be applied. The outcome is binary: you either win or lose the bet (I consider push a win). A historical probability of winning is known from your past results, 0.62 in my case. If you expect to maintain this rate, you can calculate the probability of hitting all three, two, one, and zero bets when you make three picks using the binomial distribution formula! Like it or not, you are rolling a weighted dice three times!

Here is the formula for calculating the probability of each specific outcome. Let “n” be the number of bets, “k” be the number of wins, probability of success be Ps then the probability of “k” wins out of “n” bets can be calculated using this formula:

n! / ((n-k)! k!) * (Ps) ^ k * (1-Ps) ^ (n-k)

For my example of 3 picks and 0.62% of success rate:

3! / ((3-k)! k!) * (0.62) ^ k * (0.32) ^ (3-k)

And here are the results:

# of winsprobability
3 wins23.8%
2 wins43.8%
1 win26.8%
0 wins5.5%

Two win outcome is the most probably, no wonder I am seeing it the most! I have hit zero win days, but those are rare, and that fits in line with a 5% according to the probabilities above. I should be seeing three-win days more frequently than I am but I am extremely excited at the moment to know what is expected.

I was happily going through binomial distributions topic for two days, searching for real world examples, and only after a repeated push to find one did I realize that I am living this binomial distribution outcome almost every day for the last three months!

I can now only answer the question if I should stick to three-game bets or increase this further. Let’s set the n=4, what do we get? Here is the new table:

# of winsprobability
4 wins14.7%
3 wins36.2%
2 wins33.3%
1 win13.6%
0 wins2%

In aggregate, I will end up positive for the day only if I win three or four times out of the four picks. Combined probability of that is 50.9%. Compare that with the three-bet approach, where the likelihood of coming out positive for the day is combined probabilities of hitting two and three right, which comes out to 67.6%! Should I increase the number of games I bet? The math says: no.

This realization was hugely helpful in explaining the results I was seeing. It also convinces me to stick with the current approach of three bets. I am still not clear if my 62% hit rate is blind luck or if it’s something sustainable, so I will continue to probe further as I learn and run this experiment.

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